This can be proved using the equivalent of the division algorithm for polynomials over a field.
A simple proof of this using induction on the degree ofcan be found on p.28 of D.J.H. Garling, A Course in Galois Theory (1986), Cambridge University Press.
Note that this applies to polynomials over a field only. This wont work with rings in general, e.g.and in . However, if is an integral domain, we can apply the division algorithm to polynomials over its field of fractions .
Letbe a polynomial of degree in . Let be a root of Considering as a polynomial in , we can write where and . Then .
Thus. If is another root of distinct from , write where and . Then as .
Thus. Continuing this way, we find that if are all the distinct roots of , then we have for some . It follows that cannot have more than roots in since has degree and so cannot be factorized into more than linear factors in . And since a root of polynomial in is also a root of a polynomial in (Gausss lemma), our proof is complete.
Ifis not an integral domain, it may be possible for an th-degree polynomial in to have more than roots. For instance, as a polynomial over , the quadratic polynomial actually has 4 roots, namely 1, 3, 5, 7. Thats because is not an integral domain and so we cannot construct its field of fractions.
Thanks for the reason why x^2 + 7=0 mod 8 can have 4 solutions. I didn't know that.
Last edited by bobbym (2009-08-28 12:19:02)
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.